BSTJ 60: 9. November 1981: Detecting the Occurrence of an Event by FM Through Noise. (Benes, V.E.)

Copyright © 1981 American Telephone and Telegraph Company

The Bell System Technical Journal

Vol. 60, No. 9, November 1981

Printed in U.S.A.

Detecting the Occurrence of an Event by FM
Through Noise

By V. E. BENES

(Manuscript received March 26, 1981)

The occurrence of an event at a random time r is signaled through
white noise by an fm signal whose modulation h(t — r) is a causal
pulse triggered at r. Nonlinear filtering is used to find exact expres-
sions for the chance that t > t, and the expectation of r, each
conditioned on the observed noisy fm signal over (0, t). The former
quantity can be used to minimize the probability of error in guess-
ing — from the observations over (0, t) — whether t has occurred by t.

I. INTRODUCTION

The theory of frequency modulation has always been beset by
analytical difficulties, and nowhere have these been more in evidence
than in the area of optimal demodulation of noisy fm signals. Recent
advances in nonlinear filtering, however, make it possible to solve
certain problems of detection and estimation quite explicitly. We
report on such a class of problems here.

The basic problem setup is this: an event of interest occurs at a
random time t. Its occurrence is signaled by sending a pulse of shape
/*(•), starting at t; that is, we send

,, x (o t<r

S{t) = {h(t-r) t>r>

where h(-) is some causal, integrable pulse. The signal s(t) is trans-
mitted by fm; the waveform is

cos

+ ut+ s(u)du

for a carrier frequency to and initial phase 6. In transmission this wave
suffers the degradation of having white noise added to it; thus, we
observe a signal y, defined by

2227

dy, = cos

+ cot+ \ s(u)du

o

dt + dbi ,

with b t a Brownian motion independent of t. We would like to construct
a nonlinear filter acting causally on y, to estimate optimally at each
time t whether t < t or not, and if so, by how much. This filter will be
obtained by solving the nonlinear filtering problem of determining the
conditional probability

p (t) =P{t>*|v s ,0<s<O
and the conditional density (u = distance back from t to t)
Pi (t, u) = P{red(t - tt)|*, < s<S *}, < u < t .

Such a filter (po, Pi) represents a summary, without loss, of all the
information in the "past" a{ v s , < s < £} that is relevant to whether
t occurred by time t, and if so, how far back. In particular, the filter
(Po, p\) yields least-squares estimates of t, by integration over u,
according to the formula

. uf(u)du
E{r\y,,0<s<t} =p (t) \_ F(t) + (t - u)pi(t t u)dtt ,

where F is the a priori distribution of t, and /" = F' its density. The
first term predicts where t will be, on the average, when it has not yet
occurred by t, the second "postdicts" t when it has already happened
by time t. Indeed, the first term is E{rl T>t \ y Si < s < t) and the second
isE{rl T ^\y s ,0<s^t).

II. NOTATIONS

Let x t be the process 1 T <, so that

(0 if the event has not occurred by time t.
1 if the event occurred by time t.

Then with X t = Jo x s ds, the signal s(t) can be written as

s(0 = h(t- s)dx x = < Q t<j

and the fm signal as

cos[0 + cot + H(X t )] ,

where H = /o h(s)ds.

III. FILTERING EQUATIONS

Our approach is Bayesian: foreknowledge of distr {t} is used to
calculate the conditional probabilities p () and pi. We assume for sim-

2228 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1 981

plicity, and with only slight loss of generality, that t has a known a
priori distribution F with a differentiable density /. The "rate" at
which t is occurring during (t, t + h), given that it has not yet
happened, is just

We assume at first that the phase 6 is known at the receiver. Then, the
filtering or Zakai equations for unnormalized versions p () and p\ of p
andpi, respectively, are just

dpo = \(t)p dt + cos(8 + ut)p dyt ,

dpi

dp\ = h cos[0 + ut + H(u)]pidy, ,

du

with initial conditions po(0) = 1,

lim pi(t, u)du =

HO

and boundary condition p\(t, 0) = po(t)\(t).

It can be seen that since the process x t is transient in character these
equations are coupled in one direction only: pi depends on p via the
boundary condition, but p in no way depends on pi. Thus, it will be
possible to solve for po first, and then for pi. We first transform the
problem into one without stochastic differentials. This is done by the
now familiar device 1 of looking for a solution of the form

po(t) = exp[;y,cos(0 + wt)]q (t)

pi(t, u) =exp{.y,cos[0 + ut + H(u)])q x (t, u), 0< t<u,

where q and q\ are differentiable functions, though not necessarily
C l . This form for p and p } indicates that the rough or martingale
dependence of these functions on y( • ) is confined to the exponent as
shown, while their dependence on y(-) via qo and <7i is of a much
smoother integrated form, as will be seen.

Applying Ito's formula to the postulated form, with quadratic vari-
ation d{y)i = dt since the observation process is a translation of the
Wiener process, we find these nonstochastic pdes for q and q\\

qo = qoi -Mt) + uyfiin(6 + at) - - cos 2 (0 + at) j, q (0) = 1

— = - — + qx (y,[u + h(u)]sin[6 + u>t + H(u)]
dt du \

- - cos 2 [6 + ut + H{u)]Y < u < t.

NOISE DEMODULATION 2229

The boundary condition is

qi(t,Q) = qo(t)\(t).

The first equation is an ode solvable as

q (t)=expl- X(s)ds+ [cov s sin(0 + cos) -- cos 2 (0 + cos)]ds J

= [1 - F(0]exp( [uysinid + cos) - - cos 2 (0 + cos)]tfs J .
The second is a first-order pde solvable by characteristics as

qi (t, u) = A(t - u)expl {cov s sin[0 + cos + His - t + u)]

- - cos 2 [0 + cos + H(s - t + u)]

+ yMs - t + u)sin[0 + cos + H(s - t + u)])ds J,

where A(-) is an arbitrary function. To obtain A we let u[0, and we
use h(s - t + u) = and H{s - t + u) = for s < t - u to find

1

t

2i

q x (t, 0) = i4(£)expl [> s cosin(0 + cos) - - cos (0 + cos)]cfe I,

= qo(t)X(t),
by the boundary condition. Thus, A{t) = f(t), and we obtain

qi(t, u) = fit - «)exp( [co.y s sin(0 + cos) - - cos 2 (0 + cos)]rfs

J/-U

2

o

+ h{s - t + u)]yssin[6 + cos + His - t + u)]

- - cos 2 ([0 + cos + His - t + u)])ds).

We remark that this is the unconditional density fit - u) that t
occur at t - u, multiplied by a positive factor depending on the pulse
shape h and the observation [y s , < 8 £ t). The cos 2 integral can be
evaluated explicitly, leading to some simplification, and to approximate
formulas for large carrier frequencies. 2

The normalization

Poit) + Piit, u)du = \
Jo

2230 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1 981

is achieved by dividing each of p and pi by

po(t) + p\(t, u)du,
Jo

where

po(t) = [1 - F(t)]expl y,cos(0 + ut) + [coy s sin(0 + us)

- -cos 2 (0 + us)]ds

p x (t, u) = f(t- u)exp

y,cos[6 + ut + H(u)~\

["" 1

+ [coy,sin(0 + us) - - cos 2 (0 + us)]ds

Jo 2

[u + h{s - t + u)]yssin[0 + us + H(s - t + u)]

+

H-u

- - cos 2 [0 + us + H{s -t+u)] \ds

If, as is likely, the phase 6 is not known at the receiver, then it must
be integrated out in both p and pi prior to normalization, a process
that mars the relatively neat formulas obtained for p and pi for 9
known. With uniform over (—it, it) and independent of t, familiar
Bessel function approximations again arise. 2

IV. HAS t OCCURRED YET? THE OPTIMAL GUESS

In the kind of system under study here, a task of primary interest is
to guess at t whether t has happened yet. Such a guess is represented
mathematically by a random process v,, taking the value 1 for a
decision that t has not occurred, and a value for a decision that it
has, and adapted to the past observations p{y s , < s < t). The
probability of error is just

P{r<t&. v, = 1} + P{t > t & m = 0},
which can be written as

E1 T>I (1 - v,) + EVt(l - l T>/ )
= El T>l - 2El T>l Vt + Evt
= £(1 T >, - v,) 2 ,
the mean square error in approximating 1 T> , by v,. Thus, the chance of

NOISE DEMODULATION 2231

■"

error is the least if v t is chosen to minimize this mean square error.
Noting that p (t) = £{l T >,|:y s , < s < t), we can write this mean
square error as

E{p (t) - 2p (t)v t + vt)
and conclude that a niinimizing v t is

1
v(t) =

ifpo(t)>-
if Pott) < 2'

It follows that by watching p ( • ) we can make a best guess as to
whether t has occurred yet or not, best in the sense of minimizing the
chance of being wrong.

V. THE CONDITIONAL EXPECTATION OF r

As we observe the signal y t , we may be interested in predicting t on
the basis of the information seen so far. More precisely, since it is
possible that at t > t has already occurred, we want to simultaneously
predict and "postdict" t by calculating the two terms in

r = E{T\y s , 0<s<*}

= E{Tl T>t \y a , < s < t) + E{rl^,\y a , < s < t).

The second term is clearly given in terms of pi by

(t - u)pi(t, u)du, {u = distance back to t from t) .

Jo

We claim that the first is just

udF(u)

Mt) \-F{t) •
For with o{y 8 , < s < t) = Y for short, we have

£{t1 t >, I vo} = £{t1 t >, I y'ovr > t) | vo}

= E{E{r\y l VT > t}l T> t\yo)
= E{E(T\T>t)l T>l \y' )

= P0 (t)E{T\T>t)

since the additional v s information in Y U t >t is irrelevant to t when
it is known that t > t. That is, since x, = 1 T >, is a Markov process, all

2232 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1 981

the information a{* g , y a , < s < t) is irrelevant to {x u , u > t) when it
is known that x t = 1, i.e., t > ^.

REFERENCES

1. B. L. Rozovsky, "Stochastic Partial Differential Equations Arising in Nonlinear

Filtering Problems," Uspekhi Matem. Nauk, 27 (1972), pp. 213-4.

2. V. E. Benes, "Least Squares Estimator for Frequency-Shift Position Modulation in

White Noise," B.S.T.J., 59, No. 7 (September 1970), pp. 1289-96.

NOISE DEMODULATION 2233